Large-|z| boundedness of the energy–energy correlator (EEC)

Prove that in SU(N_c) N=4 super-Yang–Mills theory at finite coupling the energy–energy correlator EEC(z), analytically continued to complex z, satisfies the asymptotic bound lim_{|z|→∞} EEC(z)/|z|^2 = 0, thereby rigorously establishing the polynomial boundedness needed to justify the Froissart–Gribov inversion formula for EEC multipoles.

Background

In the derivation of a dispersive (Froissart–Gribov) inversion formula for the EEC multipoles, the authors require assumptions about analyticity in the complex z-plane and polynomial boundedness at large |z|. While evidence from weak/strong coupling and OPE analyses suggests analyticity away from branch points at z=0 and z=1, they lack a first-principles argument for the large-|z| behavior.

To proceed, the authors conjecture a specific asymptotic bound on EEC(z)/|z|2 that would allow dropping the arc at infinity and make the dispersive inversion rigorous. Establishing this bound would solidify the mathematical foundation of their inversion formula and clarify the analytic structure of the EEC at finite coupling.

References

However, we do not have first-principle arguments to constrain the behavior in this limit. We nevertheless conjecture that at finite coupling we have \lim_{|z| \to \infty} {\text{EEC}(z) \over |z|{2} = 0 \, .

— Conformal collider bootstrap in ${\mathcal N}=4$ SYM  (2512.10796 - Dempsey et al., 11 Dec 2025) in Section 6 (Inversion formula and large spin), around equation (EECregge)